Startseite A variation on Nθ ward continuity
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

A variation on Nθ ward continuity

  • Huseyin Cakalli EMAIL logo , Mikail Et und Hacer Şengül
Veröffentlicht/Copyright: 7. Juli 2018
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

The main purpose of this paper is to introduce the concept of strongly ideal lacunary quasi-Cauchyness of sequences of real numbers. Strongly ideal lacunary ward continuity is also investigated. Interesting results are obtained.

Acknowledgements

The authors would like to thank the referees for the careful reading and several constructive comments that have improved the presentation of the results.

References

[1] D. Burton and J. Coleman, Quasi-Cauchy sequences, Amer. Math. Monthly 117 (2010), no. 4, 328–333. 10.4169/000298910x480793Suche in Google Scholar

[2] H. Çakallı, Sequential definitions of compactness, Appl. Math. Lett. 21 (2008), no. 6, 594–598. 10.1016/j.aml.2007.07.011Suche in Google Scholar

[3] H. Çakallı, Slowly oscillating continuity, Abstr. Appl. Anal. 2008 (2008), Article ID 485706. 10.1155/2008/485706Suche in Google Scholar

[4] H. Çakallı, δ-quasi-Cauchy sequences, Math. Comput. Modelling 53 (2011), no. 1–2, 397–401. 10.1016/j.mcm.2010.09.010Suche in Google Scholar

[5] H. Çakallı, Forward continuity, J. Comput. Anal. Appl. 13 (2011), no. 2, 225–230. Suche in Google Scholar

[6] H. Çakallı, On Δ-quasi-slowly oscillating sequences, Comput. Math. Appl. 62 (2011), no. 9, 3567–3574. 10.1016/j.camwa.2011.09.004Suche in Google Scholar

[7] H. Çakallı, On G-continuity, Comput. Math. Appl. 61 (2011), no. 2, 313–318. 10.1016/j.camwa.2010.11.006Suche in Google Scholar

[8] H. Çakallı, Statistical quasi-Cauchy sequences, Math. Comput. Modelling 54 (2011), no. 5–6, 1620–1624. 10.1016/j.mcm.2011.04.037Suche in Google Scholar

[9] H. Çakallı, Statistical ward continuity, Appl. Math. Lett. 24 (2011), no. 10, 1724–1728. 10.1016/j.aml.2011.04.029Suche in Google Scholar

[10] H. Çakallı, Nθ-ward continuity, Abstr. Appl. Anal. 2012 (2012), Article ID 680456. Suche in Google Scholar

[11] H. Çakallı, A variation on ward continuity, Filomat 27 (2013), no. 8, 1545–1549. 10.2298/FIL1308545CSuche in Google Scholar

[12] H. Çakallı and M. Albayrak, New type continuities via Abel convergence, Sci. World J. 2014 (2014), Article ID 398379. 10.1155/2014/398379Suche in Google Scholar

[13] H. Çakallı and P. Das, Fuzzy compactness via summability, Appl. Math. Lett. 22 (2009), no. 11, 1665–1669. 10.1016/j.aml.2009.05.015Suche in Google Scholar

[14] H. Çakallı and B. Hazarika, Ideal quasi-Cauchy sequences, J. Inequal. Appl. 2012 (2012), Paper No. 234. 10.1186/1029-242X-2012-234Suche in Google Scholar

[15] H. Çakallı and H. Kaplan, A study on Nθ-quasi-Cauchy sequences, Abstr. Appl. Anal. 2013 (2013), Article ID 836970 Suche in Google Scholar

[16] H. Çakallı and R. F. Patterson, Functions preserving slowly oscillating double sequences, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 62 (2016), no. 2, 531–536. Suche in Google Scholar

[17] H. Çakallı and E. Savaş, Statistical convergence of double sequences in topological groups, J. Comput. Anal. Appl. 12 (2010), no. 2, 421–426. Suche in Google Scholar

[18] H. Çakallı and A. Sönmez, Slowly oscillating continuity in abstract metric spaces, Filomat 27 (2013), no. 5, 925–930. 10.2298/FIL1305925CSuche in Google Scholar

[19] H. Çakallı, A. Sönmez and C. Genç, On an equivalence of topological vector space valued cone metric spaces and metric spaces, Appl. Math. Lett. 25 (2012), no. 3, 429–433. 10.1016/j.aml.2011.09.029Suche in Google Scholar

[20] H. Çakalli, A. Sönmez and C. Gündüz Aras, λ-statistically ward continuity, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 63 (2017), no. 2, 313–321. 10.1515/aicu-2015-0016Suche in Google Scholar

[21] I. Çanak and M. Dik, New types of continuities, Abstr. Appl. Anal. 2010 (2010), Article ID 258980. 10.1155/2010/258980Suche in Google Scholar

[22] A. Caserta, G. Di Maio and L. D. R. Kočinac, Statistical convergence in function spaces, Abstr. Appl. Anal. 2011 (2011), Article ID 420419. 10.1155/2011/420419Suche in Google Scholar

[23] R. Çolak, Y. Altin and M. Mursaleen, On some sets of difference sequences of fuzzy numbers, Soft Comput. 15 (2011), no. 4, 787–793. 10.1007/s00500-010-0633-8Suche in Google Scholar

[24] J. Connor and K.-G. Grosse-Erdmann, Sequential definitions of continuity for real functions, Rocky Mountain J. Math. 33 (2003), no. 1, 93–121. 10.1216/rmjm/1181069988Suche in Google Scholar

[25] J. S. Connor, The statistical and strong p-Cesàro convergence of sequences, Analysis 8 (1988), no. 1–2, 47–63. 10.1524/anly.1988.8.12.47Suche in Google Scholar

[26] P. Das, E. Savas and S. K. Ghosal, On generalizations of certain summability methods using ideals, Appl. Math. Lett. 24 (2011), no. 9, 1509–1514. 10.1016/j.aml.2011.03.036Suche in Google Scholar

[27] D. Djurčić, L. D. R. Kočinac and M. R. Žižović, Double sequences and selections, Abstr. Appl. Anal. 2012 (2012), Article ID 497594. 10.1155/2012/497594Suche in Google Scholar

[28] M. Et, B. C. Tripathy and A. J. Dutta, On pointwise statistical convergence of order α of sequences of fuzzy mappings, Kuwait J. Sci. 41 (2014), no. 3, 17–30. 10.2298/FIL1406271ESuche in Google Scholar

[29] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), FPag241–244. 10.4064/cm-2-3-4-241-244Suche in Google Scholar

[30] A. R. Freedman, J. J. Sember and M. Raphael, Some Cesàro-type summability spaces, Proc. Lond. Math. Soc. (3) 37 (1978), no. 3, 508–520. 10.1112/plms/s3-37.3.508Suche in Google Scholar

[31] J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301–313. 10.1524/anly.1985.5.4.301Suche in Google Scholar

[32] A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002), no. 1, 129–138. 10.1216/rmjm/1030539612Suche in Google Scholar

[33] B. Hazarika, Some topological and algebraic properties of paranorm i-convergent double sequence spaces, Kuwait J. Sci. 40 (2013), no. 1, 81–92. Suche in Google Scholar

[34] L. D. R. Kočinac, Selection properties in fuzzy metric spaces, Filomat 26 (2012), no. 2, 305–312. 10.2298/FIL1202305KSuche in Google Scholar

[35] P. Kostyrko, M. Mačaj, T. Šalát and M. Sleziak, -convergence and extremal -limit points, Math. Slovaca 55 (2005), no. 4, 443–464. Suche in Google Scholar

[36] P. Kostyrko, T. Šalát and W. A. A. Wilczyński, -convergence, Real Anal. Exchange 26 (2000/01), no. 2, 669–685. 10.2307/44154069Suche in Google Scholar

[37] F. Nuray and W. H. Ruckle, Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl. 245 (2000), no. 2, 513–527. 10.1006/jmaa.2000.6778Suche in Google Scholar

[38] S. K. Pal, E. Savas and H. Cakalli, I-convergence on cone metric spaces, Sarajevo J. Math. 9(21) (2013), no. 1, 85–93. 10.5644/SJM.09.1.07Suche in Google Scholar

[39] R. F. Patterson and E. Savaş, Lacunary statistical convergence of double sequences, Math. Commun. 10 (2005), no. 1, 55–61. Suche in Google Scholar

[40] T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), no. 2, 139–150. Suche in Google Scholar

[41] T. Šalát, B. C. Tripathy and M. Ziman, On some properties of -convergence, Tatra Mt. Math. Publ. 28 (2004), 279–286. Suche in Google Scholar

[42] T. Šalát, B. C. Tripathy and M. Ziman, On -convergence field, Ital. J. Pure Appl. Math. (2005), no. 17, 45–54. Suche in Google Scholar

[43] E. Savaş, Remark on double lacunary statistical convergence of fuzzy numbers, J. Comput. Anal. Appl. 11 (2009), no. 1, 64–69. 10.1155/2009/423792Suche in Google Scholar

[44] E. Savaş, On some double lacunary sequence spaces of fuzzy numbers, Math. Comput. Appl. 15 (2010), no. 3, 439–448. 10.3390/mca15030439Suche in Google Scholar

[45] E. Savaş and P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett. 24 (2011), no. 6, 826–830. 10.1016/j.aml.2010.12.022Suche in Google Scholar

[46] A. Sonmez and H. Cakalli, Cone normed spaces and weighted means, Math. Comput. Modelling 52 (2010), no. 9–10, 1660–1666. 10.1016/j.mcm.2010.06.032Suche in Google Scholar

[47] B. C. Tripathy and A. Baruah, Lacunary statically convergent and lacunary strongly convergent generalized difference sequences of fuzzy real numbers, Kyungpook Math. J. 50 (2010), no. 4, 565–574. 10.5666/KMJ.2010.50.4.565Suche in Google Scholar

[48] B. C. Tripathy and P. C. Das, On convergence of series of fuzzy real numbers, Kuwait J. Sci. Engrg. 39 (2012), no. 1A, 57–70. Suche in Google Scholar

[49] B. C. Tripathy, B. Hazarika and B. Choudhary, Lacunary I-convergent sequences, Kyungpook Math. J. 52 (2012), no. 4, 473–482. 10.5666/KMJ.2012.52.4.473Suche in Google Scholar

[50] B. C. Tripathy and S. Mahanta, On a class of generalized lacunary difference sequence spaces defined by Orlicz functions, Acta Math. Appl. Sin. Engl. Ser. 20 (2004), no. 2, 231–238. 10.1007/s10255-004-0163-1Suche in Google Scholar

[51] B. C. Tripathy and S. Mahanta, On I-acceleration convergence of sequences, J. Franklin Inst. 347 (2010), no. 3, 591–598. 10.1016/j.jfranklin.2010.02.001Suche in Google Scholar

[52] R. W. Vallin, Creating slowly oscillating sequences and slowly oscillating continuous functions, Acta Math. Univ. Comenian. (N.S.) 80 (2011), no. 1, 71–78. Suche in Google Scholar

[53] T. Yurdakadim and E. Tas, Double sequences and Orlicz functions, Period. Math. Hungar. 67 (2013), no. 1, 47–54. 10.1007/s10998-013-6362-xSuche in Google Scholar

Received: 2015-06-09
Revised: 2016-10-28
Accepted: 2017-11-27
Published Online: 2018-07-07
Published in Print: 2020-06-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 27.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/gmj-2018-0037/html
Button zum nach oben scrollen